How do they integrate this exponential?

Question

Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an exponential in it, I am not getting -1/(jn2pi). How did he get it.

Answer 1

Don't forget that $e^{-j2n\pi}=\color{blue}1$ ! $$\int_0^1 te^{at}dt=(\frac ta-\frac1{a^2})e^{at}\big|_0^1=(\frac1a-\frac1{a^2})\color{blue}1-(\frac0a-\frac1{a^2})1=\frac1a.$$